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Cell Mechanics Prof. M.S. Ju Dept. of mechanical Engineering National Cheng Kung University Tainan Outline I. Introduction & biological basis for cell mechanics (1) II. Experimental measurements of intracellular mechanics(2) III. Continuum elastic or viscoelastic models for the cell(4) IV. Models of cytoskeletal mechanics based on tensegrity (6) V. Active cellular protrusion: continuum theories and models (10) From: Mofrad & Kamm’s 1. Introduction Cells - basic functional units of life; numerous components with distinct mechanical characteristics (Prof. Tseng’s) Cell mechanics Macroscopic events: cell shape, motility, adhesion, deformation Microscopic events: sense mechanical signals & transduce them to a cascade of biochemical signals & leading to biological response Goals Describe and evaluate mechanical properties of cell and cellular structure Mechanical interaction between cell & their environment Recent focus Rheology of cytoskeleton and reconstituted gels of major cytoskeletal compopnents: actin filaments, intermediate filaments, microtubules, and their cross-linking proteins Mechanical interaction of cell with surroundings and how the interaction causes changes in morphology and biological signal that lead to functional adaptation or pathological condition Computational models Finite element – based continuum models for cell deformation Actin filament-based model for cell motility Experimental techniques Mechanical perturbation of cell by imposing deformation/force vs static/dynamic response Theories for mechanics of living cells simple elastic, viscoelastic or poro-viscoelastic continuum Porous gel or soft glassy material Tensegrity (tension integrity) network incorporating discrete structural elements (compression bearing) Test condition, length scale Objectives Providing background information on environment surrounding a cell Molecular building blocks used to impart structural strength to the cell Importance of cell mechanics in biological function (mechanobiology) 1-2 Role of cell mechanics in biological function Focused primary on eukaryotic cells of animals Exception – red blood cell or erythrocyte which do not have nucleus Living cells have time-varying properties Changing entities with capability to alter mechanical properties in response to external stimuli. Maintenance of cell shape Function of cell depends on its shape Shape is maintained through structural stiffness Examples: biconcave erythrocyte, spherical leucocytes – roll along vascular endothelium before adhere and migrate into tissue Neuronal cells extend long processes to conduct signal Airway epithelial cells covered with a bed of cilia, finger-like cell extensions; propel mucus along airway of lung Internal structure along with cell membrane provide structural integrity to maintain needed shape to accomplish its function Cell Migration Cell migrates during : Development – organism grows its various parts Wound repair – cells from undamaged tissue migrate into the wound & renew tissue Combating infection – cells of immune system transmigrate from vascular system across vessel wall into infected tissue Stages of cell migration Protrusion: extension of cell at leading edge in movement direction Adhesion of protrusion to surrounding substrate or matrix Contraction of cell that transmits a force from protrusions at leading edge to cell body, pull it forward Release of attachment at the rear Note: cell senses biochemical cues- gradient of chemotactic agents and physical environment, e.g. variations in stiffness of substrate Passive stiffness & active contractility Mechanosensing Many cells are able to sense & respond to externally applied forces The responses changes in membrane channel activity up- or down – regulation of gene expression alterations in protein synthesis altered cell morphology Example: hair cells of inner ear Hair cells Stereocilia extend from apical surface to form bundles As stereocilia bundles move in response to fluid oscillations in cochlea, tension in tip link increases, opening an ion channel to initiate electrochemical response. Ca++ Other mechanisms Conformational changes in intracellular proteins due to transmission of external forces to cell interior, leading to changes in reaction rates through a change in binding affinity Changes in viscosity of cell membrane, altering rate of diffusion of trans-membrane proteins & their rates Direct transmission of force to nucleus and to chromatin contained inside, affecting expression of specific genes. Force experienced by endothelial Glycocalyx: 醣外被 Stress responses and role of mechanical forces in disease Physical forces are instrumental in remodeling of tissues Bone remodeling/modeling, Wolff’s law Cells can sense and respond to mechanical stimulus Atherosclerosis, arthritis and pulmonary hypertension Active Cell Contraction Vascular smooth muscle cells, cardiac myocytes & skeletal muscle cells can generate force Common force generation mechanism: molecular motor comprised of actin and myosin (sacomere) Other cells contains contractile machinery for maintaining resting level of cell tension, changing cell shape or cell migration. Cardiac myocytes in culture striation Structural anatomy of a cell Cells are biologically active & their structure reflects or responds to physical environment. (primary distinction to inert materials) Thermal fluctuations need to be considered. Affects biochemical processes that lead to Intracellular remodeling Elastic characteristics of membrane Filaments of cytoskeleton Cells do not constitute structural elements of tissue Mechanical stiffness of resident cells has less contribution to modulus of tissue Deformation dictated by surrounding matrix Collagen & hydroxyapatite in bone Collagen & protoglycan w. high negative charge in cartilage In muscle, contractile force and modulus of tissue are dominated by cellular activity. In arterial wall or pulmonary airways, collagen and elastin filaments in extracellular matrix balance bulk of stress. In cardiac tissue, myocytes constitute large fraction of tissue volume and responsible for stresses & deformations of myocardium that are time-varying Extracellular matrix & its attachment to cells Force in vivo transmitted to cell via extracellular matrix (ECM) which shares load-supporting function. Cell membrane receptors’ extracellular domains bind to various proteins of ECM. Integrins to fibronectin, vitronectin, collagen and laminin Intracellular domains bind to cytoskeleton directly or indirectly. Other adhesion molecules bind to ECM, base membrane, neighboring cells or cells suspended in flowing blood. Adhesion molecules Transmission of force to cytoskeleton and role of lipid bilayer Cell membrane – thin lipid bilayer consisted of mix of phospholipids, glycolipids, cholesterol and transmembrane proteins (50% w.) Phospholipids are amphipathic Integral membrane proteins – ion channels, pathway for transmembrane signaling, structural bridge- direct adhesion between cytoskeleton & extracellular matrix. Membrane – barrier isolating cell interior from extracellular environment & maintaining biochemical condition. Bilayer contribute less to overall stiffness of cell. Fluid mosaic model (Singer & Nicolson 1972): 2-D fluid within which integral membrane proteins diffuse. Thickness ~ 6 nm, area modulus ~ 0.1-1.0 N/m (lipid bilayer) 0.45N/m (RBC), rupture strength 0.01-0.02 N/m for RBC & lipid bilayer Membrane bending rigidity: 2-4x10-19 Nm for RBC, 1-2x10-18 for neutrophils Shear modulus: 10-6 Ns/m for RBC, negligible for pure lipid bilayers Intracellular Structures Intracellular structures influence material properties of cell: cortex, projections, internal structure (nucleus, actin filaments, microtubules, intermediate filaments) Cortex: dense structure adjacent to membrane, Examples RBC, filamentous protein, spectrin, shape rigidity Epithelial cells, microvili in intestine; cilia in lung Nucleus: not well-known mechanical properties; separate contribution of nuclear envelope, and nucleoplasm. Migrating cells: leading edge sends out protrusions (lamellipodia or filopodia) rich in actin & highly cross-linked. Dynamics of actin polymerization & depolymerization Active contraction of network due to actin-myosin interactions *Dr. Lin’s PPT Actin filaments Form by polymerization of globular, monomeric actin (G-actin) into twisted strand of filamentous actin (Factin) 7-9 nm f with barbed end & pointed end ATP can bind to barbed end allows for monomer addition & filament growth Depolymerization occurs at pointed end Filament growth regulated by ionic concentrations, capping, binding, branching, and severing proteins. ATP (adenosine triphosphate)腺膘呤核苷三磷酸 From actin filaments two structures Stress fiber: tertiary structures, fiber bundle 3D lattice-like network can be formed throughactinbinding proteins (ABP) APB Examples: fimbrin & a-actinin, connect filaments into a 3D space filling matrix (gel) with filaments jointed nearly at right angle. Actin constitutes from 1-10% of all proteins in most cells, higher in muscle cells (primary structural component) Table 1-2 elastic properties Microtubules Second major constitutent of cytoskeleton. Polymerized filaments constructed from monomers of a- & b-tubulin in helical arrangement (55 kDa) Hollow cylinder, OD 25nm, high bending stiffness, persistent length 6mm, Young’s modulus similar to actin Long slender structures: cilia & flagella Network for transporting chromosomes during cell division Microtubules are highly dynamic, constant polymerization & depolymerization, half-life only few minutes Asymmetric growth, Intermediate Filaments A family of proteins > 50 members Common structure: central a helical domain of >300 residues form coil; dimers assembled into tetramers; forming protofilaments; bundle into rope-like structure, 8 protofilaments with persistent length = 1 um Long-term stability & high resistance to solubility in salts Form without GTP or ATP hydrolysis Intermediate filaments labeled with fluorescent marker can be used as fiducial markers for study the strain field within the cells High concentration in entire cell Linking Proteins Actin, microtubules & intermediate filaments are associated with cytoskeleton Linking proteins within cytoskeleton affect strength & integrity of resulting matrix. Cell membrane, nuclear membrane & all organelles & other intracellular bodies also affect mechanical response of a cell. Complexity of intracellular structure Example: focal adhesion complex Forces transmitted from extracellular matrix (fibronectin) via integral membrane adhesion receptors (a- and b-integrins), membrane associated proteins, actin-binding proteins to cytoskeleton Active contraction is fundamental feature of cytoskeleton; muscle and most cells contain contractile machinery Force measured in resting fibroblasts, stress in focal adhesion 5 Kpa! 1-3 Overview Full spectrum of views on current approaches to modeling cell mechanics. Diversity of background: biophysics, bioengineering and physical chemistry. Diversity of approaches: finite-element methods (continuum mechanics), cross-linked polymer network (cytoskeleton), soft glassy materials & gels. Static, instantaneous nature of structure and dynamic nature due to polymerization and biological processes. II. Experimental Measurements of Intracellular Mechanics Novel methods to viscoelasticity of soft materials and new theories; apply pN forces & measure nm displacement Methods: optical trap, magnetic beads, glass needle, atomic force microscope Thermal motion of refractive particles, vescicles, submicron-beads Challenge: relate experimental and theoretical results derived from molecular scale to measurement on macroscopic scale Discrepancy between nano-scale rheology & bulk measurement 2-1 Introduction Cell mechanical processes: locomotion, secretion, cell division Plant cells & bacteria: hard cell wall Animal cells: soft membrane & cytoskeleton (internal protein network) Cytoplasm: glassy, bio-solid, both viscous and elastic characteristics Measurement methods Forces to which cells are exposed in biological context Large range of stress to different tissues Cytoskeleton structures: response passively to provide strength and sensing external forces & their cellular responses Factors: molecular sensors, signaling pathways, rheological parameters Cartilage stress ~ 20MPa, chondrocytes regulate glycosaminoglycans(GAG) & others Bone & osteocytes respond to similar level; Endothelial cells ~ shear stress < 1 Pa, wide range of morphological & transcriptional changes Neutrophils activate in response to smaller shear stress Magnitude, geometry & time course of mechanical perturbations are critical Tendons, skeletal muscles experience or generate uni-axial force & deformation Blood vessel lining cells experienced shear stress due to flow; respond to changes in stress rather than magnitude Above cells & epithelial cells in lung experience large-areadilation forces, both magnitude & temporal characteristics are critical 2-2 Macrorheology, diffusion & sedimentation methods Challenges: small size of cell, heterogeneous structure of cell interior, active remodeling of cytoplasm (constitutive changes, response to external force) Stronger mechanical stimulus yield more biochemical reaction Small strain(<10%) for linear response and large strain for nonlinear response Whole cell aggregates Employ standard rheology instruments to measure stress/strain relationship on macroscopic sample containing many cells (single type) Muscle fibers: actin/myosin based fibers arranged in parallel & attached longitudinally Excellent agreement between single molecule measurement of force-elongation relation with macroscopic compliance measurement of muscle fibers sedimented samples of a single cell type, assumption: deformation is related to cell interior rather than sliding of cells past each other. Melanoma (黑色瘤(皮膚癌))cell, dictyostelium cell: single actin-binding protein mutations Disadvantages: need to verify how cells attach to each other & contribution of cell membrane or extracellular matrix. Sedimentation by particles To explore variations of viscoelastic properties within a cell, probes with size smaller than inhomogeneity are used. Diffusion or sedimentation of intracellular granules with high specific gravity in cytoplasm are observed. Large cells containing colored or retractile particles. (1950s) Record rate of falling of starch grains within a bean cell & compare with same particle in fluids of known densities & viscosities. Note viscosity of fluid can be measured by the viscometers introduced in Chapter 1. 8 mPa s for cytoplasmic viscosity Similar to falling-sphere method in macroscopic rheometry (Rockwell et al. 1984) 2 g ( - ) r v 9 2 v: velocity of particle r: particle radius g: gravity constant : viscosity of cytoplasm : density of particle : density of cytoplasm By injecting a small droplet of inert oil into a large cell (muscle fiber) and observe rising rate of the droplet, 29 mPa.s Organelles can be made to sediment by gravitational force in a centrifuge Oocyte, amoebas, slime molds have cytoplasmic viscosities 2 ~ 20 mPa.s Diffusion First centrifuging a large cell such as sea urchin egg or amoeba then monitor displacement of a single particle of radius r in one direction x(t); 4 mPa.s (cP) four times of water Sedimentation force 100~5000 times of gravity; sufficient for intracellular organelles to get concentrated while cell remains intact. Stokes-Einstein relation: kB T x (t ) 3 r 2 < > ensemble average kB Boltzmann constant T absolute temperature Three important features of intracellular material properties: Apparent viscosity of protoplasm depended strongly on flow rates Viscous flow of internal organelles could be measured only deeper inside the cell Cellular viscosity is temperature dependent 2-3 Mechanical Indentation of Cell Surface Glass micro-needles Cell poker Atomic Force Microscopy Glass micro-needle To apply force large enough to deform the cell but no damage of cell Pull on individual cultured neurons, point forces for initiating neurite extension Calibration of clamped wire needle by hanging a weight at free end and the deflection y F L3 y ( L) , 3 EI I r4 4 The wire needle is used to calibrate a thinner glass and repeated till a glass microneedle that can detect nN or smaller force Cell poker Cell suspended in fluid Vertical glass needle attached to wire needle that coupled to a piezoelectric actuator Vertical displacements of both end of wire are measured optically x = difference of displacement F= k x Resolutions: displacement < 100nm, force < 10 nN Typical force vs. displacement curve Elasticity and plastic (?) deformation of cell Tip is smaller than cell, local viscoelasticity can be probed at different regions of cell findings Large difference in relative stiffness over different areas High degree of softening when actin-filamentdisorganization drugs like cytochalasin were applied Apparent stiffness increased as amplitude of indentation increased Discussion Hertz model is not valid if cell thickness is not much greater than indentation depth. Cell cytoskeleton is not isotropic homogeneous material Force exerted on a cell initiate biochemical and other active reaction (Daily et al. 1984) Atomic Force Microscopy Imaging by scanning a sharp microscopy tip over a surface while recording tip deflection simultaneously (Scanning Probe Microscope) Deflection time course - surface profile Three modes: contact mode, tapping mode, jumping mode & others AFM indentation can be used to probe mechanical properties at selected locations Better spatial & force resolution than cell poking Atomic Force Microscope System (Agilent 5500 ) 57 Contact mode z 位移感測 A B - C D d vertical A B C D A C - B D dlaterial A B C D 回饋系統 接觸模式 User defined Servo and scanning parameters y o Error User defined Set-point deflection Controller Electronics Laser Feedback signal for PZT scanner + - Instantaneous Deflection signal AB CD PZT Scanner for XYZ movement Cantilever and tip Sample laser X Y u v Power Amplifier z Z PZT contact indented q A A d Sample Substrate F kd d z - d cos q d z A’ q d Indentation of semi-space elastic solid (Hertz model) Sphere indenter Fs (2.4) Conical indenter Fc 3 4 E R d 3 (1 - 2 ) E tana d 2 2 2 (1 - ) (2.5) Regular Pyramid (Bilodeau 1992) tan a E d 2 F 0.7453 1 - 2 Soma of PC-12 cell topography deflection Semi-contact mode (tapping mode) PZT-xyz Excitation source u(t) 未受力 k b Sample ξ=0 +ξ Fext(ξ) A Ao m Liquid 受外力 t t -ξ0 • 作用小 6 Amplitude (nm) 0 10 15 15 20 0 0 10 m 15 20 10 20 m 200 400 600 800 20 0 10 m -5 10 m 20 20 0 (B) m 200 400 600 800 Topography (nm) -5 0(nm) 5 Amplitude 0 A 15 0 10 m 20 15 0 10 m 20 0 20 10 20 m 0 10 20 m 10 12 16 18 20 16 18 E (kPa) 20 10 18 m 0 20 20 Phase delay (degree) 0 10 5 5 15 m 10 20 10 20 15 B 16 5 5 15 m m 10 20 往左掃描 5 15 20 往左掃描:16.52±0.50 kPa 誤差為1.27% 往右掃描:16.31±0.53 kPa 5 15 E (kPa) 0 10 6 8 (degree) 10 12 Phase delay 0 0 m 0 10 85 10 12 14 Amplitude (nm) Vertical position (m) Line A-B (μm) 0 10 m m 2 4 6 Topography (nm) 5 20 10 E (kPa) m 6 8 20 0 5 20 10 15 0 E 18 (kPa) 20 0 0 5 18 5 0 16 20 B 0 15 16 15.5 10 20 10 m 16 A 0 10 20 20 0 5 2015 0 18 20 m m 10 m 18 16 10 6 8 (degree) 10 Phase delay E (kPa) 0 Scan0 direction:Left Scan5 direction:Right 5 Phase delay (degree) 10 0 10 5 15 0 8 m m m 5 Topography (nm) 5 15 16.5 20 2 -6Amplitude -4 -2 (nm) 0 2 0 m Elastic modulus (kPa) 200 400 600 800 Topography (nm) 0 10 17 10 20 0 16 0 5 17.5 -2 m (A) -4 10 20 15 0 10 m E (kPa) 往右掃描 0 10 5 15 m 0 -6 m 18 200 400 600 800 m 0 m • 側向力小 10 20 20 0 15 10 m 20 63 Limitations: manipulating through accessible surface of cells can not measure elastic moduli well inside cell Sharp tip (50 nm f) and Hertz model to test various types of cells (1999). Cells are inhomogeneous, anisotropic, active biosolids. Cell periphery are crucial for cell motility, too thin to apply standard Hertz model Spatial inhomogeneity of cells (stress fibers, microtubules) may be filtered by large probe Different mechanical properties between cell center and cell periphery Quasi-static or low frequency measurements may induce active cellular responses in addition to passive material properties. High frequency tapping mode are also used but cell dynamically stiffened by the rapidly oscillating tip. Polystyrene beads of proper radius to contact cell (Mahaffy 2004, 2000) Bead radius & inhomogeneity relationship Zero-frequency shear modulus: 1~2 kPa for fibroblast Complex modulus at 50-300Hz were measured and evaluated with extended Hertz model. Hydrodynamic drag on rest of cantilever is not easy to compensate. Hertz model has been modified to account for finite sample thickness and boundary condition on substrate. Applied to thin lamellipodia of cells: Young’s modulus of 1~2 kPa for fibroblasts 2.4 Mechanical Tension applied to cell membrane Pulling of cell membrane by controlled suction within micropipette to measure viscosity & elastic response of cells (1954) Red Blood Cell with continuous viscoelastic protein network lining membrane. Advantages: suspended in solution or attached to a surface Also for viscoelasticity of leukocytes and monocytes(Chien 1984, Dong 1988, Richelme 2000) Micropipette experiment of cells Model: liquid drop with cortical tension Measured quantities: cortical tension in membrane, cytoplasmic viscosity, cell elasticity for leukocyte under some conditions Elastic body model (f~2.1) 2 L p P E f 3 Rp (2.6) Liquid-like flow of cells, viscosity R p p Rp d Lp m (1 - ) R dt (2.7), m 9 Shearing & Compression L Attach cells at top and bottom to glass surface moved w.r.t. each other in compression, extension, or shear. Fibroblasts adhered to glass surface coated with adhesion proteins (fibronectin) on rigid plate Second flexible plate on top surface PZT motor drives rigid plate to impose strain of cell and measure deflection of flexible plate to calculate stress. Imaging internal structure at same time Fluid Flow Cell like vascular endothelial or osteocyte within bone matrix are regularly exposed to fluid stresses Cell senses shear stress, crucial for regulatory processes In vascular endothelial, mechanosensing control production of protective ECM In bone, mechansensing is basis of bone repair and adaptive structure processes In vitro shearing of osteocytes in parallel plate flow chamber Fluid flow system to stimulate mechanosensitive bone cells • Monolayer of osteocytes • Controlled shear stress • NO as func. of flow rate Strain filed within cell measured by endogeneous fluorescent vimentin (Helmke 2001, 2003) Inhomogeneous strain field, focused on localized area Sites of mechanosensing Large strain location if distortion of sensing element is required Focused stress, small strain but high force Numerical simulations applied to cell & fluid passing over it. Combined FEM & CFD to model flow across surface of adhering cell & calculate shear stress in different spots. Without knowing intracellular material inhomogeneities (linear elastic & isotropic) Also applied to cells manipulated by AFM, magnetic beads, substrate stretching, useful methods! z Optical traps (Laser Tweezers) Laser beam forcused through high numerical aperture r objective lens to 3D trap refractive particles (silica or latex beads) Force acting on bead: approximation for small particles (Gaussian focus, size < laser wavelength) Gradient forces a a 'ia " complex polarity I 0 : laser intensity w0 : beam radius at focus 2 Fgr -a ' I 0 r 2 2 0 w e w4 ( z ) - w (z) (2.8) z0 w02 2 : Rayleigh range, k m : wave vector m m m : wave length in medium with refractive index n m 2r 2 2 w(z) w 0 2r 2 w 1 2r - w 2 ( z ) F -a ' I 0 z e z w 4 ( z ) w6 ( z ) a g 4 0 2 0 z 1 : beam radius near focus z0 (2.9) 2 0 0: vacuum wavelength of light NA D D: diameter of beam waist Scattering forces 2r 2 2 2 2 - 2 w w0 r z - z0 k m 1 e w (z) Fsa a " I 0 2 2 2 w ( z ) 2 z 2 z02 z0 w ( z ) 2 0 Fsr a " I 0 2 0 w km r e 2 w ( z ) R( z ) - 2r 2 w2 ( z ) 2 (2.10) (2.11) Stable trapping: gradient force wins over the trapping force; Stability depends on geometry and properties of trapped particle & surrounding medium. Particle size & relative index of refraction n=np/nm Scattering force ~ d6, gradient force ~ d3 (A) Origin of Fscat and Fgrad for high index sphere displaced from TEM00 beam axis. Ashkin A PNAS 1997;94:4853-4860 ©1997 by The National Academy of Sciences of the USA (A) Geometry of levitation trap. Ashkin A PNAS 1997;94:4853-4860 ©1997 by The National Academy of Sciences of the USA Working principle Trappable bead attached to surface of cell & deform cell locally; Advantages: no mechanical access, bead of um size yield high resolution, Disadvantages: force < 100pN << micropipetts, AFM; local heating; Using interferometric method, great accuracy force & displacement (sub nm, sub pN) Fast detection: 10us useful for cell viscoelasticity Possible to differentiate active, motor driven response & passive viscoelasticity! Examples Human RBC, 2D shear modulus 2.5~200 uN/m Magnetic Methods Magnetic fields to apply forces and/or torques to ferromagnetic or paramagnetic particles adhered to cells. (large force, no open surface) Difficult to establish homogeneous field gradients; limited video rates. Freundlich & Seifriz (1922), insert nickel or magnetite particle to large cell, magnetic field gradient generated by electromagnet, impose force on bead whose displacement measured by microscope; force calibrated by using fluid with known viscosity. Magnetic manipulation system Crick & Hughes (1950) modification: Magnetize particle with large magnetic field & use a smaller probing magnetic field at different angle to twist particles. Phagocytic cells engulf magnetic particles to avoid damage to cell Motion of embedded beads depends on probe size; bead might interact with & stick to cytoskeleton cause active motion! How beads are coupled to the network ? Summary Experimental methods for mechanical properties of cells Macrorheology, diffusion, sedimentation Mechanical indentation Mechanical tension Shearing & compression Optical traps Magnetic methods 3. Continuum elastic or viscoelastic models for the cell Continuum model: when smallest length scale is larger than dimensions of microstructure Whole-cell deformation: 1~2 order Micropipette aspiration of erythrocytes or neutrophils Magnetocytometry: bead size & deformation >> mesh size of cytoskeleton network Continuum model: no constraints on isotropy, inhomogeneity of properties Constitutive law - stress-strain relation; linear elastic to viscoelastic a coarse-grain approach replacing contribution of cytoskeleton’s stress discrete fibers with averaged constitutive law Review of elastic and viscoelastic continuum multicompartment description of the cell Finite-element-based 2D & 3D models of the cell with compartments (membrane, actin cortex, cytoskeleton & nucleus) Contrasting computational results against experimental data obtained from different methods. Purpose of continuum models Analyze experiments probing single cell mechanics Evaluate force level sensed by various parts of cell in vitro or in vivo Compare theoretical and computational predictions against experimental observations and deduce mechanical properties of cell Examples: magnetocytometry, micropipette aspiration, microindentation, AFM RBS Aspiration Neutrophil indentation (Left) & passage (right) Effect of fluid shear AFM indentation Simulation of optical tweezers test on RBC Comparison of continuum models with corresponding experiments can help to distinguish active biological responses of the cell from passive mechanical deformation. Evaluate strain & stress under biological conditions: endothelium, neutrophils, fibroblasts… Principles of continuum models Continuum model provides displacement, strain and stress fields induced in cell, given initial geometry, mechanical properties, and boundary conditions (displacements or force on cell surface) Laws of continuum mechanics are used to solved for distribution of stress and deformation Discretize cell volume into small elements using FEM techniques. Assume body and inertial force are negligible Conservation of linear momentum σ 0 where s is the Cauchy stress tensor Boundary conditions Traction boundary (surface force) Displacement boundary (displacement =0) Material mechanical properties Cytoskeleton: difficult to model, active/passive, Nucleus: stiffer & more viscous, elastic, 18Pa~10kPa Cellular membrane: thin (5nm), viscoelastic time constant 10us Cortex: stiffer than rest of cytoskeleton, RBC, cortical tension Examples of studied cells Blood cells: leukocytes & erythrocytes Adherent cells: fibroblasts, epithelial cells & endothelial cells Geometry of model Deformed shapes FE model of cell monolayer pulled by magnetic cytometry FEM simulation of cell contact sites on basal cell surface (focal adhesion) Limitations of continuum model Aim at passive dynamics of cells Not account for active biology: deformation and stresses experienced as a consequence of biochemical response of cell to mechanical loading. May isolate phenomena involving active biology from passive mechanical response of the cell Continuum model might simulate active processes through time-dependent properties or residual strains linked to biological processes Lack of description of cytoskeletal fibers, not applicable for micro-manipulation of cell with a probe <= cytoskeletal mesh (0.1-1um) Exclude Brownian motions due to thermal fluctuations of cytoskeleton, network node & cell motility Limited number of time constants to characterize cell’s viscoelastic behavior Need continuous spectrum of time scales (QLV?) Conclusion Continuum mechanical models useful in exploiting & interpreting experimental results from probing single cell or cell monolayer Identify stress & strain patterns induced within the cell or mechanical properties of cell compartments. Predict forces experienced within cells in vivo and form hypotheses on how cell might sense & transduce forces into shape change or gene expression Viscous and viscoelastic models is needed